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Recently, I read the Chapter 8 of Johnstone's "Topos theory" and got interested in the homotopy and cohomology theory of Grothendieck toposes. So I'm looking for the textbooks expanding these subjects, and it seems that it will be treated in the Volume 3 of "Sketches of an Elephant".

However, unfortunately, Volume 3 has not yet been published for 10 years. So, my question is

  • Volume 3 will not be published? Did he give up writing?
  • Is a part of Volume 3 available? Is there a preprint?
  • Does there exist a textbook which include those subjects? (Homotopy theory of toposes, Model structures for sheaves, Torsors and non-abelian cohomology, etc..)
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  • $\begingroup$ It is very unlikely that Johnstone will release the material electronically - he has not authorised an electronic copy of volumes 1 and 2 and has said that he won't. $\endgroup$ Commented Apr 7, 2014 at 5:47

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Volume 3 is still in preparation. Some sections of Part F have been completed; you can try to ask the author for a copy if you're interested. I do not know if much progress has been made on Part E, but here are some alternatives:

  • The original reference for the homotopy theory of toposes is [Artin and Mazur, Étale homotopy], and some aspects are discussed in [Moerdijk, Classifying spaces and classifying topoi].
  • Model structures on presheaf toposes are the subject of Cisinski's thesis, [Les préfaisceaux comme modèles des types d'homotopie].
  • There doesn't seem to be a coherent reference for model structures for sheaves of simplicial sets at this time.
  • The original reference for non-abelian cohomology is probably [Giraud, Cohomologie non abélienne].
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  • $\begingroup$ @Zhen Lin, any further updates? I think I heard the SDG part was done, or close to. $\endgroup$ Commented May 12, 2015 at 3:12
  • $\begingroup$ @DavidRoberts I had heard that there is a draft version of the SDG chapter but I haven't seen it myself. $\endgroup$
    – Zhen Lin
    Commented May 12, 2015 at 7:14
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    $\begingroup$ Any updates on when (if) Volume III will be published? $\endgroup$ Commented Feb 16, 2019 at 0:26
  • $\begingroup$ @Rachmaninoff I am asking this same question but now... I know Jonhstone had a bad experience with COVID-19 and I do not know if he is still active. If someone knew something, it would be great to know! I still hope there will be a 3rd volumen. $\endgroup$
    – Hvjurthuk
    Commented Dec 9, 2022 at 11:53

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